LAPACK
3.4.2
LAPACK: Linear Algebra PACKage

Functions/Subroutines  
subroutine  cptcon (N, D, E, ANORM, RCOND, RWORK, INFO) 
CPTCON  
subroutine  cpteqr (COMPZ, N, D, E, Z, LDZ, WORK, INFO) 
CPTEQR  
subroutine  cptrfs (UPLO, N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO) 
CPTRFS  
subroutine  cpttrf (N, D, E, INFO) 
CPTTRF  
subroutine  cpttrs (UPLO, N, NRHS, D, E, B, LDB, INFO) 
CPTTRS  
subroutine  cptts2 (IUPLO, N, NRHS, D, E, B, LDB) 
CPTTS2 solves a tridiagonal system of the form AX=B using the L D LH factorization computed by spttrf. 
This is the group of complex computational functions for PT matrices
subroutine cptcon  (  integer  N, 
real, dimension( * )  D,  
complex, dimension( * )  E,  
real  ANORM,  
real  RCOND,  
real, dimension( * )  RWORK,  
integer  INFO  
) 
CPTCON
Download CPTCON + dependencies [TGZ] [ZIP] [TXT]CPTCON computes the reciprocal of the condition number (in the 1norm) of a complex Hermitian positive definite tridiagonal matrix using the factorization A = L*D*L**H or A = U**H*D*U computed by CPTTRF. Norm(inv(A)) is computed by a direct method, and the reciprocal of the condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
[in]  N  N is INTEGER The order of the matrix A. N >= 0. 
[in]  D  D is REAL array, dimension (N) The n diagonal elements of the diagonal matrix D from the factorization of A, as computed by CPTTRF. 
[in]  E  E is COMPLEX array, dimension (N1) The (n1) offdiagonal elements of the unit bidiagonal factor U or L from the factorization of A, as computed by CPTTRF. 
[in]  ANORM  ANORM is REAL The 1norm of the original matrix A. 
[out]  RCOND  RCOND is REAL The reciprocal of the condition number of the matrix A, computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is the 1norm of inv(A) computed in this routine. 
[out]  RWORK  RWORK is REAL array, dimension (N) 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value 
The method used is described in Nicholas J. Higham, "Efficient Algorithms for Computing the Condition Number of a Tridiagonal Matrix", SIAM J. Sci. Stat. Comput., Vol. 7, No. 1, January 1986.
Definition at line 120 of file cptcon.f.
subroutine cpteqr  (  character  COMPZ, 
integer  N,  
real, dimension( * )  D,  
real, dimension( * )  E,  
complex, dimension( ldz, * )  Z,  
integer  LDZ,  
real, dimension( * )  WORK,  
integer  INFO  
) 
CPTEQR
Download CPTEQR + dependencies [TGZ] [ZIP] [TXT]CPTEQR computes all eigenvalues and, optionally, eigenvectors of a symmetric positive definite tridiagonal matrix by first factoring the matrix using SPTTRF and then calling CBDSQR to compute the singular values of the bidiagonal factor. This routine computes the eigenvalues of the positive definite tridiagonal matrix to high relative accuracy. This means that if the eigenvalues range over many orders of magnitude in size, then the small eigenvalues and corresponding eigenvectors will be computed more accurately than, for example, with the standard QR method. The eigenvectors of a full or band positive definite Hermitian matrix can also be found if CHETRD, CHPTRD, or CHBTRD has been used to reduce this matrix to tridiagonal form. (The reduction to tridiagonal form, however, may preclude the possibility of obtaining high relative accuracy in the small eigenvalues of the original matrix, if these eigenvalues range over many orders of magnitude.)
[in]  COMPZ  COMPZ is CHARACTER*1 = 'N': Compute eigenvalues only. = 'V': Compute eigenvectors of original Hermitian matrix also. Array Z contains the unitary matrix used to reduce the original matrix to tridiagonal form. = 'I': Compute eigenvectors of tridiagonal matrix also. 
[in]  N  N is INTEGER The order of the matrix. N >= 0. 
[in,out]  D  D is REAL array, dimension (N) On entry, the n diagonal elements of the tridiagonal matrix. On normal exit, D contains the eigenvalues, in descending order. 
[in,out]  E  E is REAL array, dimension (N1) On entry, the (n1) subdiagonal elements of the tridiagonal matrix. On exit, E has been destroyed. 
[in,out]  Z  Z is COMPLEX array, dimension (LDZ, N) On entry, if COMPZ = 'V', the unitary matrix used in the reduction to tridiagonal form. On exit, if COMPZ = 'V', the orthonormal eigenvectors of the original Hermitian matrix; if COMPZ = 'I', the orthonormal eigenvectors of the tridiagonal matrix. If INFO > 0 on exit, Z contains the eigenvectors associated with only the stored eigenvalues. If COMPZ = 'N', then Z is not referenced. 
[in]  LDZ  LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if COMPZ = 'V' or 'I', LDZ >= max(1,N). 
[out]  WORK  WORK is REAL array, dimension (4*N) 
[out]  INFO  INFO is INTEGER = 0: successful exit. < 0: if INFO = i, the ith argument had an illegal value. > 0: if INFO = i, and i is: <= N the Cholesky factorization of the matrix could not be performed because the ith principal minor was not positive definite. > N the SVD algorithm failed to converge; if INFO = N+i, i offdiagonal elements of the bidiagonal factor did not converge to zero. 
Definition at line 146 of file cpteqr.f.
subroutine cptrfs  (  character  UPLO, 
integer  N,  
integer  NRHS,  
real, dimension( * )  D,  
complex, dimension( * )  E,  
real, dimension( * )  DF,  
complex, dimension( * )  EF,  
complex, dimension( ldb, * )  B,  
integer  LDB,  
complex, dimension( ldx, * )  X,  
integer  LDX,  
real, dimension( * )  FERR,  
real, dimension( * )  BERR,  
complex, dimension( * )  WORK,  
real, dimension( * )  RWORK,  
integer  INFO  
) 
CPTRFS
Download CPTRFS + dependencies [TGZ] [ZIP] [TXT]CPTRFS improves the computed solution to a system of linear equations when the coefficient matrix is Hermitian positive definite and tridiagonal, and provides error bounds and backward error estimates for the solution.
[in]  UPLO  UPLO is CHARACTER*1 Specifies whether the superdiagonal or the subdiagonal of the tridiagonal matrix A is stored and the form of the factorization: = 'U': E is the superdiagonal of A, and A = U**H*D*U; = 'L': E is the subdiagonal of A, and A = L*D*L**H. (The two forms are equivalent if A is real.) 
[in]  N  N is INTEGER The order of the matrix A. N >= 0. 
[in]  NRHS  NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. 
[in]  D  D is REAL array, dimension (N) The n real diagonal elements of the tridiagonal matrix A. 
[in]  E  E is COMPLEX array, dimension (N1) The (n1) offdiagonal elements of the tridiagonal matrix A (see UPLO). 
[in]  DF  DF is REAL array, dimension (N) The n diagonal elements of the diagonal matrix D from the factorization computed by CPTTRF. 
[in]  EF  EF is COMPLEX array, dimension (N1) The (n1) offdiagonal elements of the unit bidiagonal factor U or L from the factorization computed by CPTTRF (see UPLO). 
[in]  B  B is COMPLEX array, dimension (LDB,NRHS) The right hand side matrix B. 
[in]  LDB  LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). 
[in,out]  X  X is COMPLEX array, dimension (LDX,NRHS) On entry, the solution matrix X, as computed by CPTTRS. On exit, the improved solution matrix X. 
[in]  LDX  LDX is INTEGER The leading dimension of the array X. LDX >= max(1,N). 
[out]  FERR  FERR is REAL array, dimension (NRHS) The forward error bound for each solution vector X(j) (the jth column of the solution matrix X). If XTRUE is the true solution corresponding to X(j), FERR(j) is an estimated upper bound for the magnitude of the largest element in (X(j)  XTRUE) divided by the magnitude of the largest element in X(j). 
[out]  BERR  BERR is REAL array, dimension (NRHS) The componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any element of A or B that makes X(j) an exact solution). 
[out]  WORK  WORK is COMPLEX array, dimension (N) 
[out]  RWORK  RWORK is REAL array, dimension (N) 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value 
ITMAX is the maximum number of steps of iterative refinement.
Definition at line 183 of file cptrfs.f.
subroutine cpttrf  (  integer  N, 
real, dimension( * )  D,  
complex, dimension( * )  E,  
integer  INFO  
) 
CPTTRF
Download CPTTRF + dependencies [TGZ] [ZIP] [TXT]CPTTRF computes the L*D*L**H factorization of a complex Hermitian positive definite tridiagonal matrix A. The factorization may also be regarded as having the form A = U**H *D*U.
[in]  N  N is INTEGER The order of the matrix A. N >= 0. 
[in,out]  D  D is REAL array, dimension (N) On entry, the n diagonal elements of the tridiagonal matrix A. On exit, the n diagonal elements of the diagonal matrix D from the L*D*L**H factorization of A. 
[in,out]  E  E is COMPLEX array, dimension (N1) On entry, the (n1) subdiagonal elements of the tridiagonal matrix A. On exit, the (n1) subdiagonal elements of the unit bidiagonal factor L from the L*D*L**H factorization of A. E can also be regarded as the superdiagonal of the unit bidiagonal factor U from the U**H *D*U factorization of A. 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = k, the kth argument had an illegal value > 0: if INFO = k, the leading minor of order k is not positive definite; if k < N, the factorization could not be completed, while if k = N, the factorization was completed, but D(N) <= 0. 
Definition at line 93 of file cpttrf.f.
subroutine cpttrs  (  character  UPLO, 
integer  N,  
integer  NRHS,  
real, dimension( * )  D,  
complex, dimension( * )  E,  
complex, dimension( ldb, * )  B,  
integer  LDB,  
integer  INFO  
) 
CPTTRS
Download CPTTRS + dependencies [TGZ] [ZIP] [TXT]CPTTRS solves a tridiagonal system of the form A * X = B using the factorization A = U**H*D*U or A = L*D*L**H computed by CPTTRF. D is a diagonal matrix specified in the vector D, U (or L) is a unit bidiagonal matrix whose superdiagonal (subdiagonal) is specified in the vector E, and X and B are N by NRHS matrices.
[in]  UPLO  UPLO is CHARACTER*1 Specifies the form of the factorization and whether the vector E is the superdiagonal of the upper bidiagonal factor U or the subdiagonal of the lower bidiagonal factor L. = 'U': A = U**H*D*U, E is the superdiagonal of U = 'L': A = L*D*L**H, E is the subdiagonal of L 
[in]  N  N is INTEGER The order of the tridiagonal matrix A. N >= 0. 
[in]  NRHS  NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. 
[in]  D  D is REAL array, dimension (N) The n diagonal elements of the diagonal matrix D from the factorization A = U**H*D*U or A = L*D*L**H. 
[in]  E  E is COMPLEX array, dimension (N1) If UPLO = 'U', the (n1) superdiagonal elements of the unit bidiagonal factor U from the factorization A = U**H*D*U. If UPLO = 'L', the (n1) subdiagonal elements of the unit bidiagonal factor L from the factorization A = L*D*L**H. 
[in,out]  B  B is REAL array, dimension (LDB,NRHS) On entry, the right hand side vectors B for the system of linear equations. On exit, the solution vectors, X. 
[in]  LDB  LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = k, the kth argument had an illegal value 
Definition at line 122 of file cpttrs.f.
subroutine cptts2  (  integer  IUPLO, 
integer  N,  
integer  NRHS,  
real, dimension( * )  D,  
complex, dimension( * )  E,  
complex, dimension( ldb, * )  B,  
integer  LDB  
) 
CPTTS2 solves a tridiagonal system of the form AX=B using the L D LH factorization computed by spttrf.
Download CPTTS2 + dependencies [TGZ] [ZIP] [TXT]CPTTS2 solves a tridiagonal system of the form A * X = B using the factorization A = U**H*D*U or A = L*D*L**H computed by CPTTRF. D is a diagonal matrix specified in the vector D, U (or L) is a unit bidiagonal matrix whose superdiagonal (subdiagonal) is specified in the vector E, and X and B are N by NRHS matrices.
[in]  IUPLO  IUPLO is INTEGER Specifies the form of the factorization and whether the vector E is the superdiagonal of the upper bidiagonal factor U or the subdiagonal of the lower bidiagonal factor L. = 1: A = U**H *D*U, E is the superdiagonal of U = 0: A = L*D*L**H, E is the subdiagonal of L 
[in]  N  N is INTEGER The order of the tridiagonal matrix A. N >= 0. 
[in]  NRHS  NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. 
[in]  D  D is REAL array, dimension (N) The n diagonal elements of the diagonal matrix D from the factorization A = U**H *D*U or A = L*D*L**H. 
[in]  E  E is COMPLEX array, dimension (N1) If IUPLO = 1, the (n1) superdiagonal elements of the unit bidiagonal factor U from the factorization A = U**H*D*U. If IUPLO = 0, the (n1) subdiagonal elements of the unit bidiagonal factor L from the factorization A = L*D*L**H. 
[in,out]  B  B is REAL array, dimension (LDB,NRHS) On entry, the right hand side vectors B for the system of linear equations. On exit, the solution vectors, X. 
[in]  LDB  LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). 
Definition at line 114 of file cptts2.f.